Globally asymptotic stability of a stochastic mutualism model with saturated response

A two-species stochastic mutualism model with saturated response is proposed and investigated in this paper. We demonstrate that there exists a unique positive solution to the model for any positive initial value. Under some conditions, we show that the stochastic model is globally asymptotically stable. Finally, we work out some figures to illustrate our results.

Globally Asymptotic Stability of a Delayed Integro-Differential Equation With Nonlocal Diffusion

We study a population model with nonlocal diòusion, which is a delayed integro-diòerential equation with double nonlinearity and two integrable kernels. By comparison method and analytical technique, we obtain globally asymptotic stability of the zero solution and the positive equilibrium. The results obtained reveal that the globally asymptotic stability only depends on the property of nonlinearity. As an application, we discuss an example for a population model with age structure.

Existence and Globally Asymptotic Stability of Equilibrium Solution for Fractional-Order Hybrid BAM Neural Networks with Distributed Delays and Impulses

This paper investigates the existence and globally asymptotic stability of equilibrium solution for Riemann-Liouville fractional-order hybrid BAM neural networks with distributed delays and impulses. The factors of such network systems including the distributed delays, impulsive effects, and two different fractional-order derivatives between the U-layer and V-layer are taken into account synchronously. Based on the contraction mapping principle, the sufficient conditions are derived to ensure the existence and uniqueness of the equilibrium solution for such network systems. By constructing a novel Lyapunov functional composed of fractional integral and definite integral terms, the globally asymptotic stability criteria of the equilibrium solution are obtained, which are dependent on the order of fractional derivative and network parameters. The advantage of our constructed method is that one may directly calculate integer-order derivative of the Lyapunov functional. A numerical example is also presented to show the validity and feasibility of the theoretical results.

Globally asymptotic stability of a predator–prey model with stage structure incorporating prey refuge

This paper focuses on the stabilities of the equilibria to a predator–prey model with stage structure incorporating prey refuge. By analyzing the characteristic functions, we obtain that the equilibria of the model are locally stable when some suitable conditions are being satisfied. According to the comparison theorem and iteration technique, the globally asymptotic stability of the positive equilibrium is discussed. And, the sufficient conditions of the global stability to the trivial equilibrium and the boundary equilibrium are derived. The study shows that the prey refuge will enhance the density of the prey species, and it will decrease the density of predator species. Finally, some numerical simulations are carried out to show the efficiency of our main results.

An Approach of Tracking Control for Chaotic Systems

Combining the ergodicity of chaos and the Jacobian matrix, we design a general tracking controller for continuous and discrete chaotic systems. The control scheme has the ability to track a bounded reference signal. We prove its globally asymptotic stability and extend it to generalized projective synchronization. Numerical simulations verify the effectiveness of the proposed scheme.

Globally Asymptotic Stability of Stochastic Nonlinear Systems with Time-Varying Delays via Output Feedback Control

We address the problem of globally asymptotic stability for a class of stochastic nonlinear systems with time-varying delays. By the backstepping method and Lyapunov theory, we design a linear output feedback controller recursively based on the observable linearization for a class of stochastic nonlinear systems with time-varying delays to guarantee that the closed-loop system is globally asymptotically stable in probability. In particular, we extend the deterministic nonlinear system to stochastic nonlinear systems with time-varying delays. Finally, an example and its simulations are given to illustrate the theoretical results.